A212760 Number of (w,x,y,z) with all terms in {0,...,n}, w even, and x = y + z.

1, 3, 12, 20, 45, 63, 112, 144, 225, 275, 396, 468, 637, 735, 960, 1088, 1377, 1539, 1900, 2100, 2541, 2783, 3312, 3600, 4225, 4563, 5292, 5684, 6525, 6975, 7936, 8448, 9537, 10115, 11340, 11988, 13357, 14079, 15600, 16400, 18081, 18963, 20812, 21780, 23805

Offset: 0

Author

Clark Kimberling, May 29 2012

Keywords

• nonn
• easy

Comments

A signed version is A122576.

For a guide to related sequences, see A211795.

Partial sums of the positive elements of A129194. - Omar E. Pol, Dec 28 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A211795.

Cf. A001318, A260706.

Programs

Haskell

a212760 = a260706 . fromInteger . a001318 . (+ 1)
-- Reinhard Zumkeller, Nov 17 2015

Magma

[(n+1)*(n+2)*(2*n+3+(-1)^n)/8 : n in [0..50]]; // Wesley Ivan Hurt, Jul 22 2014

Maple

A212760:=n->(n+1)*(n+2)*(2*n+3+(-1)^n)/8: seq(A212760(n), n=0..50); # Wesley Ivan Hurt, Jul 22 2014

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && x == y + z, s++],
{w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
Map[t[#] &, Range[0, 50]]  (* A212760 *)
Table[(n + 1) (n + 2) (2 n + 3 + (-1)^n)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Jul 22 2014 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 + x)^3 (1 - x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 23 2014 *)

Formula

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).

G.f.: ( 1+2*x+6*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).

a(n) = (n+1)*(n+2)*(2*n+3+(-1)^n)/8. - Wesley Ivan Hurt, Jul 22 2014

a(n) = A260706(A001318(n+1)). - Reinhard Zumkeller, Nov 17 2015

a(n) = Sum_{i=1..n+1} numerator(i^2/2). - Wesley Ivan Hurt, Feb 26 2017